Estimation and prediction of the Burr type XII distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.     

Estimation with the generalized exponential distribution based on record values and inter-record times

The maximum likelihood and Bayesian approaches have been considered for the two-parameter generalized exponential distribution based on record values with the number of trials following the record values (inter-record times). The maximum likelihood estimates are obtained under the inverse sampling and the random sampling schemes. It is shown that the maximum likelihood estimator of the shape parameter converges in mean square to the true value when the scale parameter is known. The Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo methods due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The confidence intervals for the parameters are constructed based on asymptotic and Bayesian methods. The Bayes and the maximum likelihood estimators are compared in terms of the estimated risk by the Monte Carlo simulations. The comparison of the estimators based on the record values and the record values with their corresponding inter-record times are performed by using Monte Carlo simulations.     

Bayesian analysis for the two-parameter Pareto distribution based on record values and times

Doostparast and Balakrishnan (Pareto record-based analysis, Statistics, under review) recently developed optimal confidence intervals as well as uniformly most powerful tests for one- and two-sided hypotheses concerning shape and scale parameters, for the two-parameter Pareto distribution based on record data. In this paper, on the basis of record values and inter-record times from the two-parameter Pareto distribution, maximum-likelihood and Bayes estimators as well as credible regions are developed for the two parameters of the Pareto distribution. For illustrative purposes, a data set on annual wages of a sample of production-line workers in a large industrial firm is analysed using the proposed procedures.     

Evidential inference based on record data and inter-record times

Science looks to statistics for an objective measure of the strength of evidence in a given body of observations. In this paper, we shall use a criterion defined as a combination of probabilities of weak and strong misleading evidence to do the comparison between only record values and the same number of record values and inter-record times. Also, a simulation is presented to illustrate the results.     

Estimation based on progressive first-failure censoring from exponentiated exponential distribution

In this paper, point and interval estimations for the parameters of the exponentiated exponential (EE) distribution are studied based on progressive first-failure-censored data. The Bayes estimates are computed based on squared error and Linex loss functions and using Markov Chain Monte Carlo (MCMC) algorithm. Also, based on this censoring scheme, approximate confidence intervals for the parameters of EE distribution are developed. Monte Carlo simulation study is carried out to compare the performances of the different methods by computing the estimated risks (ERs), as well as Akaike's information criteria (AIC) and Bayesian information criteria (BIC) of the estimates. Finally, a real data set is introduced and analyzed using EE and Weibull distributions. A comparison is carried out between the mentioned models based on the corresponding Kolmogorov–Smirnov (K–S) test statistic to emphasize that the EE model fits the data with the same efficiency as the other model. Point and interval estimation of all parameters are studied based on this real data set as illustrative example.     

Bayesian estimation and prediction based on lognormal record values

In this paper we consider the problems of estimation and prediction when observed data from a lognormal distribution are based on lower record values and lower record values with inter-record times. We compute maximum likelihood estimates and asymptotic confidence intervals for model parameters. We also obtain Bayes estimates and the highest posterior density (HPD) intervals using noninformative and informative priors under square error and LINEX loss functions. Furthermore, for the problem of Bayesian prediction under one-sample and two-sample framework, we obtain predictive estimates and the associated predictive equal-tail and HPD intervals. Finally for illustration purpose a real data set is analyzed and simulation study is conducted to compare the methods of estimation and prediction.     

Estimation with the lognormal distribution on the basis of records

Record scheme is a method to reduce the total time on test of an experiment. In this scheme, items are sequentially observed and only values smaller than all previous ones are recorded. In some situations, when the experiments are time-consuming and sometimes the items are lost during the experiment, the record scheme dominates the usual random sample scheme [M. Doostparast and N. Balakrishnan, Optimal sample size for record data and associated cost analysis for exponential distribution, J. Statist. Comput. Simul. 80(12) (2010), pp. 1389–1401]. Estimation of the mean of an exponential distribution based on record data has been treated by Samaniego and Whitaker [On estimating population characteristics from record breaking observations I. Parametric results, Naval Res. Logist. Q. 33 (1986), pp. 531–543] and Doostparast [A note on estimation based on record data, Metrika 69 (2009), pp. 69–80]. The lognormal distribution is used in a wide range of applications when the multiplicative scale is appropriate and the log-transformation removes the skew and brings about symmetry of the data distribution [N.T. Longford, Inference with the lognormal distribution, J. Statist. Plann. Inference 139 (2009), pp. 2329–2340]. In this paper, point estimates as well as confidence intervals for the unknown parameters are obtained. This will also be addressed by the Bayesian point of view. To carry out the performance of the estimators obtained, a simulation study is conducted. For illustration proposes, a real data set, due to Lawless [Statistical Models and Methods for Lifetime Data, 2nd ed., John Wiley & Sons, New York, 2003], is analysed using the procedures obtained.     

Bayes estimation and prediction of the two-parameter gamma distribution

In this article, the Bayes estimates of two-parameter gamma distribution are considered. It is well known that the Bayes estimators of the two-parameter gamma distribution do not have compact form. In this paper, it is assumed that the scale parameter has a gamma prior and the shape parameter has any log-concave prior, and they are independently distributed. Under the above priors, we use Gibbs sampling technique to generate samples from the posterior density function. Based on the generated samples, we can compute the Bayes estimates of the unknown parameters and can also construct HPD credible intervals. We also compute the approximate Bayes estimates using Lindley's approximation under the assumption of gamma priors of the shape parameter. Monte Carlo simulations are performed to compare the performances of the Bayes estimators with the classical estimators. One data analysis is performed for illustrative purposes. We further discuss the Bayesian prediction of future observation based on the observed sample and it is seen that the Gibbs sampling technique can be used quite effectively for estimating the posterior predictive density and also for constructing predictive intervals of the order statistics from the future sample.     

Statistical inference based on generalized Lindley record values

This paper addresses the problems of frequentist and Bayesian estimation for the unknown parameters of generalized Lindley distribution based on lower record values. We first derive the exact explicit expressions for the single and product moments of lower record values, and then use these results to compute the means, variances and covariance between two lower record values. We next obtain the maximum likelihood estimators and associated asymptotic confidence intervals. Furthermore, we obtain Bayes estimators under the assumption of gamma priors on both the shape and the scale parameters of the generalized Lindley distribution, and associated the highest posterior density interval estimates. The Bayesian estimation is studied with respect to both symmetric (squared error) and asymmetric (linear-exponential (LINEX)) loss functions. Finally, we compute Bayesian predictive estimates and predictive interval estimates for the future record values. To illustrate the findings, one real data set is analyzed, and Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation and prediction.     

Bayesian inference and prediction of order statistics for a Type-II censored Weibull distribution

This paper describes the Bayesian inference and prediction of the two-parameter Weibull distribution when the data are Type-II censored data. The aim of this paper is twofold. First we consider the Bayesian inference of the unknown parameters under different loss functions. The Bayes estimates cannot be obtained in closed form. We use Gibbs sampling procedure to draw Markov Chain Monte Carlo (MCMC) samples and it has been used to compute the Bayes estimates and also to construct symmetric credible intervals. Further we consider the Bayes prediction of the future order statistics based on the observed sample. We consider the posterior predictive density of the future observations and also construct a predictive interval with a given coverage probability. Monte Carlo simulations are performed to compare different methods and one data analysis is performed for illustration purposes.     

Bayesian inference for the randomly censored Weibull distribution

In this paper, we consider the Bayesian inference of the unknown parameters of the randomly censored Weibull distribution. A joint conjugate prior on the model parameters does not exist; we assume that the parameters have independent gamma priors. Since closed-form expressions for the Bayes estimators cannot be obtained, we use Lindley's approximation, importance sampling and Gibbs sampling techniques to obtain the approximate Bayes estimates and the corresponding credible intervals. A simulation study is performed to observe the behaviour of the proposed estimators. A real data analysis is presented for illustrative purposes.     

Estimation of parameters of bivariate normal distribution using concomitants of record values

In this paper, we discuss the concomitants of record values arising from the well-known bivariate normal distribution BVND(μ₁, μ₂,σ₁,σ₂, ρ). We have obtained the best linear unbiased estimators of μ₂ and σ₂ when ρ is known and derived some unbiased linear estimators of ρ when μ₂ and σ₂ are known, based on the concomitants of first n record values. The variances of these estimators have been obtained.     

Shortest tolerance intervals controlling both tails of the exponential distribution based on record values

This paper deals with computing shortest width tolerance intervals controlling both tails of the exponential distribution on the basis of record values. Equal-tailed and shortest tolerance factors are derived. The expected widths of these tolerance intervals are evaluated via a Monte Carlo simulation study. Finally, two illustrative examples are also included.     

Statistical inference about the shape parameter of the Weibull distribution by upper record values

Summary In this paper, we provide some pivotal quantities to test and establish confidence interval of the shape parameter on the basis of the firstn observed upper record values. Finally, we give some examples and the Monte Carlo simulation to assess the behaviors (including higher power and more shorter length of confidence interval) of these pivotal quantities for testing null hypotheses and establishing confidence interval concerning the shape parameter under the given significance level and the given confidence coefficient, respectively.     

Bayes Estimation Based on Random Censored Data for Some Life Time Models Under Symmetric and Asymmetric Loss Functions

Censored data arise naturally in a number of fields, particularly in problems of reliability and survival analysis. There are several types of censoring; in this article, we shall confine ourselves to the right randomly censoring type. Under the Bayesian framework, we study the estimation of parameters in a general framework based on the random censored observations under Linear-Exponential (LINEX) and squared error loss (SEL) functions. As a special case, Weibull model is discussed and the admissibility of estimators of parameters verified. Finally, a simulation study is conducted based on Monte Carlo (MC) method for comparing estimated risks of the estimators obtained.     

Bayesian prediction of waiting times in stochastic models

The authors show how saddlepoint techniques lead to highly accurate approximations for Bayesian predictive densities and cumulative distribution functions in stochastic model settings where the prior is tractable, but not necessarily the likelihood or the predictand distribution. They consider more specifically models involving predictions associated with waiting times for semi‐Markov processes whose distributions are indexed by an unknown parameter θ. Bayesian prediction for such processes when they are not stationary is also addressed and the inverse‐Gaussian based saddlepoint approximation of Wood, Booth & Butler (1993) is shown to accurately deal with the nonstationarity whereas the normal‐based Lugannani & Rice (1980) approximation cannot, Their methods are illustrated by predicting various waiting times associated with M/M/q and M/G/1 queues. They also discuss modifications to the matrix renewal theory needed for computing the moment generating functions that are used in the saddlepoint methods.     

Bayesian inference of three-parameter bathtub-shaped lifetime distribution

We investigate a Bayesian inference in the three-parameter bathtub-shaped lifetime distribution which is obtained by adding a power parameter to the two-parameter bathtub-shaped lifetime distribution suggested by Chen (2000). The Bayes estimators under the balanced squared error loss function are derived for three parameters. Then, we have used Lindley's and Tierney–Kadane approximations (see Lindley 1980; Tierney and Kadane 1986) for computing these Bayes estimators. In particular, we propose the explicit form of Lindley's approximation for the model with three parameters. We also give applications with a simulated data set and two real data sets to show the use of discussed computing methods. Finally, concluding remarks are mentioned.     

Estimation for the generalized half-normal distribution based on progressive type-II censoring

Based on progressively type-II censored data, the problem of estimating unknown parameters and reliability function of a two-parameter generalized half-normal distribution is considered. Maximum likelihood estimates are obtained by applying expectation-maximization algorithm. Since they do not have closed forms, approximate maximum likelihood estimators are proposed. Several Bayesian estimates with respect to different symmetric and asymmetric loss functions such as squared error, LINEX and general entropy are calculated. The Lindley approximation method is applied to determine Bayesian estimates. Monte Carlo simulations are performed to compare the performances of the different methods. Finally, one real data set is analysed.     

Bayesian estimation and prediction for Weibull model with progressive censoring

This article presents the statistical inferences on Weibull parameters with the data that are progressively type II censored. The maximum likelihood estimators are derived. For incorporation of previous information with current data, the Bayesian approach is considered. We obtain the Bayes estimators under squared error loss with a bivariate prior distribution, and derive the credible intervals for the parameters of Weibull distribution. Also, the Bayes prediction intervals for future observations are obtained in the one- and two-sample cases. The method is shown to be practical, although a computer program is required for its implementation. A numerical example is presented for illustration and some simulation study are performed.